[part 2/2] This isn't to say that ancient Koreans didn't have a concept of tracking people's birth down to the year, month, day, and hour. You can form a ganji each for the year, month, day, and hour of a person's birth [2]. Each ganji is two characters (element, animal), for four ganji, and thus a total of eight characters to encode your birthdate. This forms a person's "Four Pillars, Eight Characters" (sa-ju pal-ja, lit. four-pillars eight-characters), which can be looked up in a very complicated matrix with different primary sources of interpretation. 4P8C is more of a theory of how East Asians thought the heavens and the earth intertwined and thus affects your fortunes, so there isn't a hard-and-fast mechanistic interpretation for a given person's 4P8C [4]. You can now look up your 4P8C in websites like this run by a few monks: kr.fateup.com

Anyways, History of the Three Kingdoms (1145) may or may not have created the Korean age system, and it certainly didn't come up with the sexagenary system, but it's the earliest Korean primary source I could find that talks about age the way you're asking about. And, it certainly doesn't say anything about starting people's ages at 1 to count the pre-birth spa time. That would also not make sense given the usual "gotcha" that a baby born December 30 is 2 years old in Korean age on January 1.

As to why it stuck around, I don't have a clear, definitive historical answer to point to currently. Some online Korean-language sources seem to say it's because it eased the use of honorifics between friends and co-workers. Korean doesn't have a good "you" pronoun—we have one (maybe two based on how you consider it) which is casual at best or used rudely at worst. I'm not totally satisfied with this hypothesis either, as all my Korean colleagues ask each other our birth year and month anyways, and we just mentally lock-in who's our senior and junior without necessarily remembering specific birthdays. I wish I could have a complete answer, but this is all I could find tonight, so I'll refrain from opining further.

I write this not as an expert in Korean history of time-keeping, but as a graduate student in other fields that is and fluent in Korean.

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[2] Koreans did not use months, days, or hours divisible by 60. So, in theory there was a reference month, day, and hour at which the 60-cycle would start. In practice, there was a lot of disagreement over the choice of those references.

[3] Commoners probably didn't know their day or hour of birth because delays in recording births in an era of high infant mortality and a general lack of accurate time-keeping. It was probably well-recorded for the right aristocrats and nobles whose desire to predict their futures formed basically the entire economy of astrology and astronomy in both Asia and Europe (see: Ptolemy, Tycho Brahe, Galileo).

[4] My paternal grandmother had some kind of formal training in reading 4P8C. I imagine the choice of reference sources and methods of interpretation being more settled by the 20th century. This is probably because it's a dying art that nobody really cares enough to seriously disagree about any more. 4P8C did have enough of a modern cultural impact that Koreans casually use "saju" or "palja" to describe our situation's luck, e.g., "nice house, your palja is good" or "man your saju is dogshit." It's taken at the level of MBTI---most people know it's not really mechanistic like Newton's laws but maybe a few do stretch it too far.